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Coefficient of Variation, Variance, and Standard Deviation Explained in 2020

There are many ways to quantify variability, however, here we will focus on the most common ones: variancestandard deviation, and coefficient of variation. In the field of statistics, we typically use different formulas when working with population data and sample data.
Different formulas-variability, coefficient of variation

Sample Formulas vs Population Formulas

When we have the whole population, each data point is known so you are 100% sure of the measures we are calculating.
different formulas for population data-variability, coefficient of variation
When we take a sample of this population and compute a sample statistic, it is interpreted as an approximation of the population parameter.
A sample statistic is an approximation of the population parameter-variability, coefficient of variation
Moreover, if we extract 10 different samples from the same population, we will get 10 different measures.
10 different samples give 10 different measures-variability, coefficient of variation
Statisticians have solved the problem by adjusting the algebraic formulas for many statistics to reflect this issue. Therefore, we will explore both population and sample formulas, as they are both used.

The Mean, Median and Mode

You must be asking yourself why there are unique formulas for the meanmedian and mode. Well, actually, the sample mean is the average of the sample data points, while the population mean is the average of the population data points. As you can see in the picture below, there are two different formulas, but technically, they are computed in the same way.
sample formula vs population formula-variability, coefficient of variation
After this short clarification, it’s time to get onto variance.

Variance Formula: Sample Variance and Population Variance

Variance measures the dispersion of a set of data points around their mean value.
variance measures the dispersion of a set of data points around their mean-variability, coefficient of variation
Population variance, denoted by sigma squared, is equal to the sum of squared differences between the observed values and the population mean, divided by the total number of observations.
population variance-variability, coefficient of variation
Sample variance, on the other hand, is denoted by s squared and is equal to the sum of squared differences between observed sample values and the sample mean, divided by the number of sample observations minus 1.
Sample Variance-variability, coefficient of variation

A Closer Look at the Formula for Population Variance

When you are getting acquainted with statistics, it is hard to grasp everything right away. Therefore, let’s stop for a second to examine the formula for the population and try to clarify its meaning. The main part of the formula is its numerator, so that’s what we want to comprehend. The sum of differences between the observations and the mean, squared. So, this means that the closer a number is to the mean, the lower the result we obtain will be. And the further away from the mean it lies, the larger this difference.
higher and lower results-variability, coefficient of variation

Why do we Elevate to the Second Degree

Squaring the differences has two main purposes.
  1. First, by squaring the numbers, we always get non-negative computations. Without going too deep into the mathematics of it, it is intuitive that dispersion cannot be negative. Dispersion is about distance and distance cannot be negative.
higher and lower results squared-variability, coefficient of variation
If, on the other hand, we calculate the difference and do not elevate to the second degree, we would obtain both positive and negative values that, when summed, would cancel out, leaving us with no information about the dispersion.
  1. Second, squaring amplifies the effect of large differences. For example, if the mean is 0 and you have an observation of 100, the squared spread is 10,000!
10 squared is 100-variability, coefficient of variation

Putting the Population Formula to Use

Alright, enough dry theory. It is time for a practical example. We have a population of five observations – 1, 2, 3, 4 and 5. Let’s find its variance. We start by calculating the mean: (1 + 2 + 3 + 4 + 5) / 5 = 3.
we start by calculating the mean-variability, coefficient of variation
Then we apply the formula which we just discussed: ((1 – 3)2 + (2 – 3)2+ (3 – 3)2 + (4 – 3)+ (5 – 3)2) / 5.
observation mean, coefficient of variation
When we do the math, we get 2. So, the population variance of the data set is 2.
population variance formula, coefficient of variation

Calculating the Sample Variance

But what about the sample variance? This would only be suitable if we were told that these five observations were a sample drawn from a population. So, let’s imagine that’s the case. The sample mean is once again 3. The numerator is the same, but the denominator is going to be 4, instead of 5.
n - 1 4, coefficient of variation
This gives us a sample variance of 2.5.
sample variance, coefficient of variation

Why the Results are not the Same

To conclude the variance topic, we should interpret the result. Why is the sample variance bigger than the population variance? In the first case, we knew the population. That is, we had all the data and we calculated the variance. In the second case, we were told that 1, 2, 3, 4 and 5 was a sample, drawn from a bigger population.

The Population of the Sample

Imagine that the population of the sample were the following 9 numbers: 1, 1, 1, 2, 3, 4, 5, 5 and 5.
imaginary population, coefficient of variation
Clearly, the numbers are the same, but there is a concentration around the two extremes of the data set – 1 and 5. The variance of this population is 2.96.
Coefficient of Variation
So, our sample variance has rightfully corrected upwards in order to reflect the higher potential variability. This is the reason why there are different formulas for sample and population data.
Coefficient of Variation

Standard Deviation Formula: Sample Standard Deviation and Population Standard Deviation

While variance is a common measure of data dispersion, in most cases the figure you will obtain is pretty large. Moreover, it is hard to compare because the unit of measurement is squared. The easy fix is to calculate its square root and obtain a statistic known as standard deviation. In most analyses, standard deviation is much more meaningful than variance.

The Formulas

Similar to the variance there is also population and sample standard deviation. The formulas are: the square root of the population variance and square root of the sample variance respectively. I believe there is no need for an example of the calculation. Anyone with a calculator in their hands will be able to do the job.
standard deviation formulas, coefficient of variation

The Coefficient of Variation (CV)

The last measure which we will introduce is the coefficient of variation. It is equal to the standard deviation, divided by the mean.
coefficient of variation
Another name for the term is relative standard deviation. This is an easy way to remember its formula – it is simply the standard deviation relative to the mean.
standard deviation relative to the mean, coefficient of variation
As you probably guessed, there is a population and sample formula once again.
sample formula, coefficient of variation

Why We Need the Coefficient of Variation

So, standard deviation is the most common measure of variability for a single data set. But why do we need yet another measure such as the coefficient of variation? Well, comparing the standard deviations of two different data sets is meaningless, but comparing coefficients of variation is not.
comparing two or more datasets
Aristotle once said: “Tell me, I’ll forget. Show me, I’ll remember. Involve me, I’ll understand.”
“Tell me, I’ll forget. Show me, I’ll remember. Involve me, I’ll understand.”

Examples of Comparing Standard Deviations

To make sure you remember, here’s an example of a comparison between standard deviations. Let’s take the prices of pizza at 10 different places in New York. As you can see in the picture below, they range from 1 to 11 dollars.
Now, imagine that you only have Mexican pesos. To you, the prices will look more like 18.81 pesos to 206.91 pesos, given the exchange rate of 18.81 pesos for one dollar.
pizza price example
Let’s combine our knowledge so far and find the standard deviations and coefficients of variation of these two data sets.

Sample or Population Data

  1. First, we have to see if this is a sample or a population. Are there only 11 restaurants in New York? Of course not. This is obviously a sample drawn from all the restaurants in the city. Then we have to use the formulas for sample measures of variability.
sample or population?

Finding the Mean

  1. Second, we have to find the mean. The mean in dollars is equal to 5.5 and the mean in pesos to 103.46.
find the mean

Calculating the Sample Variance and the Standard Deviation

  1. The third step of the process is finding the sample variance. Following the formula that we went over earlier, we can obtain 10.72 dollars squared and 3793.69 pesos squared.
  2. The respective sample standard deviations are 3.27 dollars and 61.59 pesos, as shown in the picture below.
Find the sample deviation

A Few Observations

Let’s make a couple of observations. First, variance gives results in squared units, while standard deviation in original units, as shown below.
Coefficient of Variation
This is the main reason why professionals prefer to use standard deviation as the main measure of variability. It is directly interpretable. Squared dollars mean nothing, even in the field of statistics. Second, we got standard deviations of 3.27 and 61.59 for the same pizza at the same 11 restaurants in New York City. However, this seems wrong. Let’s make it right by using our last tool – the coefficient of variation.

The Advantage of the Coefficient of Variation

We can divide the standard deviations by the respective means. As you can see in the picture below, we get the two coefficients of variation.
CV= s/x
The result is the same – 0.60. Important: Notice that it is not dollars, pesos, dollars squared or pesos squared. It is just 0.60. This shows us the great advantage that the coefficient of variation gives us. Now, we can confidently say that the two data sets have the same variability, which was what we expected beforehand.
In the picture above, you can see the main advantages of the coefficient of variation. The article first appeared on: https://365datascience.com/coefficient-variation-variance-standard-deviation/

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